First slide

System of linear equations

Question

x,y,z not all zeros and the equations $x = c y + b z, y = a z + c x, z = b x + a y$ are consistent then selection among a,b,c is

Moderate

A

$a^{2} + b^{2} + c^{2} = 0$
B

$a^{2} + b^{2} + c^{2} - 2 a b c = 0$
C

$a^{2} + b^{2} + c^{2} + 2 a b c = 1$
D

$a^{2} + b^{2} + c^{2} + 2 a b c = 0$

Solution

$T h e h o m o g e n e o u s s y s t e m h a s n o n - t r i v i a l s o l u t i o n \Rightarrow Δ = 0$
$\Rightarrow |\begin{array}{l} 1 - c - b \\ c - 1 a \\ b a - 1 \end{array}| = 0 \Rightarrow (1 - a^{2}) + c (- c - a b) - b (a c + b) = 0$
$\Rightarrow 1 - a^{2} - c^{2} - a b c - a b c - b^{2} = 0$
$\Rightarrow a^{2} + b^{2} + c^{2} + 2 a b c = 1$

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